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@guy I've grasped all that, the only thing I cannot see implicitly (because I am not that experienced with these expressions) is how I would write them as functions of the quantities mentioned. The rest would be trivial.
Hello and thanks for your input ! I think that this $$\text{Var}(\hat{y}_{x_0}) = \displaystyle\frac{\sigma^2\sum x_i^2}{n\sum(x_i - \bar{x})^2} + \displaystyle\frac{\sigma^2x_0^2}{\sum(x_i - \bar{x})^2} - \displaystyle\frac{2x_0\sigma^2\bar{x}}{\sum(x_i - \bar{x})^2}$$ needs to be proven as well though.
Interesting and indeed true. I assume the exercise wants us to work under the assumption of Gaussian distributed errors regarding a fitted generalized model, but that isn't mentioned, so I guess your statement is perfect !
@a_statistician If $ Y = X\beta + \epsilon$ then $\bar{Y} = \bar{X}\beta + \epsilon$ and $\hat{Y} = X\hat{\beta} + \hat{\epsilon}$ and then it is : $$(Y-\hat Y)'(\hat Y - \bar Y) = (X\beta + \epsilon - X\hat{\beta} + \hat{\epsilon})'(X\hat{\beta} + \hat{\epsilon}- \bar{X}\beta + \epsilon)$$ Now, what ?
